:,» 


OF  THE 


UNIVERSITY  OF  CALIFORNIA. 


QIKT  OK 


ORBIT  OF  PSYCHE 


A  REVISED  FORM  OF  A  THESIS  PRESENTED  TO  THE  FACULTY  OF  THE  UNIVERSITY 

OF  MICHIGAN,  IN  PARTIAL  FULFILLMENT  OF  THE   REQUIREMENTS 

FOR   THE    DEGREE   OF   DOCTOR   OF   SCIENCE 


BY 


SIDNEY  DEAN  TOWNLEY 


OF  THE 

UNIVERSITY 

OF 


SAN     FRANCISCO 

C.  A.  MURDOCK   &    Co.,  PRINTERS 
1905 


TABLE    OF    CONTENTS. 

PAGE. 

HISTORICAL  NOTE   1 

COMPUTATION  OF  EPHEMERIS  FOR  OPPOSITION  OF  1882 1 

COMPUTATION  OF  EPHEMERIS  FOR  OPPOSITION  OF  1885 1 

DISCUSSION  OF  STAR  PLACES 2 

COLLECTION  OF  OBSERVATIONS 2 

WEIGHTS  OF  OBSERVATIONS 3 

COMPARISON  OF  OBSERVED  AND  COMPUTED  COORDINATES 4 

COMPUTATION  OF  DIFFERENTIAL  COEFFICIENTS 5 

LEAST  SQUARE  SOLUTION  OF  OBSERVATION  EQUATIONS 5 

COMPUTATION  OF  CHECK  EPHEMERIS 7 

COMPUTATION  OF  SPECIAL  PERTURBATIONS  BY  JUPITER  1870-1876 7 

COMPARISON  WITH  SCHUBERT'S  PERTURBATIONS 7 

DISCUSSION  OF  DIFFERENCES  BETWEEN  SCHUBERT'S  AND' MY  PERTURBATIONS.  8 

SPECIAL  PERTURBATIONS  FROM  1900  TO  1906 11 


THE    OliBIT    OF    PSYCHE. 


By  SIDNEY  DEAN  TOWNLEY. 


The  planet  Psyche  was1  discovered  at  Naples 
March  17,  1852,  by  Professor  'ANNIBEL  DE  GASPARis.1 
This  discovery,  as  is  the  case  in  many  others  both 
greater  and  less,  was  made  by  two  men, — the  second 
discovery  being  only  one  day  later  than  the  first.  Mr. 
HIND,  of  London,  charted  this  planet  nearly  two 
months  before  the  discovery  of  DE  GASPARIS  and  found 
it  to  be  an  asteroid  on  March  18th.- 

The  planet  was  immediately  observed  at  several  ob- 
servatories, and  soon  afterwards  the  first  elements 
were  computed  by  RUMKER.:{  Orbits  were  also  com- 
puted by  VOGEL,  SCHONFELD,  SONTAG,  and  KLINKER- 
FUES.  The  last  named  was  then  an  assistant  of  GAUSS, 
and  he  used  GAUSS'S  method  of  computing  the  orbit 
from  four  places,  which,  on  account  of  the  small  in- 
clination of  the  plane  of  the  orbit,  gave  a  much  better 
determination  of  the  orbit  than  was  obtained  by  the 
other  computers,  who  used"  only  three  observations..4 
Ir;  1855  KLINKERFUES  computed  perturbations  by 
Jupiter  and  corrected  the  elements  by  all  the  observa- 
tions made  up  to  that  time.5  Afterwards  AUWERS 
corrected  KLINKERFUES  's  elements  by  the  observations 
of  the  oppositions  of  1857  and  1858,  computing  per- 
turbations by  both  Jupiter  and  Saturn.0  About  1860 
SCHUBERT  commenced  computations  upon  this  planet 
and  corrected  KLINKERFUES 's  elements  from  the  ob- 
servations of  eight  oppositions.7  Frequent  observa- 
tions of  the  planet  were  made  between  1860  and  1870, 
and  SCHUBERT  again  corrected  the  elements  and  com- 
puted special  perturbations  by  Jupiter  from  1870 
to  1900.8 


Since  1870  but  few  observations  and  no  computa- 
tions have  been  made  upon  the  planet.  My  first 
observations  were  secured  at  the  opposition  of  1894. 
A  comparison  of  the  observed  and  computed  places 
showed  the  planet  to  be  considerably  out  of  the 
ephemeris  place,  and  it  was  deemed  advisable  to  cor- 
rect SCHUBERT'S  elements.  It  was  found  that  ob- 
servations had  been  made  at  the  oppositions  of  1879, 
1880,  1882,  1883,  and  1885.  As  there  were  no  other 
observations  until  1892  it  was  decided  to  divide  the 
work  into  two  parts,  as  follows:  First — To  correct 
SCHUBERT'S  elements  from  the  observations  of  1879, 
1880,  1882,  1883,  and  1885;  with  the  corrected  ele- 
ments, to  recompute  the  special  perturbations  by 
Jupiter  over  enough  of  the  period  from  1870  to  1900 
to  enable  one  to  determine  empirical  formulae  from 
which  to  compute  the  corrections  to  SCHUBERT'S  per- 
turbations at  any  epoch.  Second — To  again  correct 
the  elements  from  the  observations  1890  to  1900, 
and  with  these  Anally  corrected  elements,  to  compute 
perturbations  from  1900  on.0 

Computation  of  Ephemeris.  For  each  of  «the  op- 
positions under  consideration,  those  of  1879,  1880, 
1882,  1883,  and  1885,  an  ephemeris  of  Psyche,  com- 
puted from  SCHUBERT'S  elements  was  published  in 
the  Berliner  Astronomischcs  Jahfbuch.  At  two  of 
the  oppositions,  however,  observations  were  made  after 
the  expiration  of  the  ephemeris,  and  it  became  neces- 
sary to  compute  extensions  for  each.  The  results  of 
the  computations  are  as  follows : — 


Berlin   M.  T. 

a 

5                              log    A             Aberr.  T. 

1882     March 

13d 

12h  00m  OOs 

10h 

9m    4s.23 

4-  11°  55'    50".8             0.358562 

18m  57s 

14 

12 

00 

00 

10 

8 

25 

.66 

4-12       0       5  .3             0.359788 

19       0 

15 

12 

00 

00 

10 

7 

47 

.96 

4-12       4     14  .1             0.361061 

19       3 

16 

12 

00 

00 

10 

7 

11 

.18 

4-12       8     16  .9             0.362379 

19       6 

17 

12 

00 

00 

10 

6 

35 

.33 

4-12     12     13  .7             0.363742 

19     10 

18 

12 

00 

00 

10 

6 

0 

.47 

4-  12     16       4  .3             0.365148 

19     14 

19 

12 

00 

00 

10 

5 

26 

.61 

4-  12     19     48  .5             0.366597 

19     18 

1882     March 

20 

12 

00 

00 

10 

4 

53 

.79 

4-  12     23     26  .1              0.368087 

19     22 

1885     Dec. 

1 

12 

1 

37 

4 

15 

1 

.87 

4-  16     19     40  .9             0.221533 

13     49 

4 

11 

47 

6 

4 

12 

22 

.42 

4-16     14     18  .0             0.223776 

13     53 

9 

11 

23 

5 

4 

8 

8 

.61 

4-10       6     31  .9             0.228924 

14       3 

23 

10 

17 

43 

3 

58 

32 

.75 

4-  15     54     56  .6             0.251592 

14     48 

1885     Dec. 

30 

9 

46 

31 

3 

55 

24 

.38 

4-15     55     41   .0             0.266562 

15     20 

1  A.   N.   Bd.  34,  147, 

5  Ibid. 

38,  181. 

!(  The  plan  of  this  work  was  adopted 

while 

the  author  was 

-  Ibid.  34,  163. 

8  ibid. 

50,  375. 

a  student  at  the  University  of  Michigan. 

It  is 

now  recognized 

3  Ibid.  34,  189. 

7  Ibid. 

57,  323. 

that  the  plan  is  open  to  criticism.     In  the  first 

place,  it  would 

4  Ibid.  35,  17. 

8  Ibid. 

75,  209. 

have  been  much  better  to  have  used  all  of 

the  oppositions  avail- 

157974 


Observations.  Three  different  types  of  instru- 
ments were  used  in  making  observations  upon  this 
planet, — the  equatorial,  the  heliometer,  and  the 
meridian  circle.  Fifteen  comparison  stars  were  used 
in  the  equatorial  and  heliometer  observations.  Their 
places  have  been  carefully  investigated  and  the  results 
are  incorporated  in  the  following  tabulation.  STRUVE'S 
precessional  constants  have  been  used  in  bringing 
up  the  catalogue  places.  Where  possible  reductions 
to  the  A.  G.  system  have  been  applied.  Proper  motions 
have  been  applied  in  only  one  case — star  number 
eight,  The  values  used  are  -f  Os.0071  and  —  0".03, 


and  were  obtained  from  the  Poulkova  Catalogues. 
The  positions  of  all  the  other  stars  were  examined  for 
evidence  of  proper  motion,  but  none  of  a  positive 
nature  was  found.  In  assigning  weights  to  the  various 
catalogue  positions  I  have  been  guided  largely  by  the 
table  published  by  Professor  NEWCOMB  in  his  Cata- 
logue of  Fundamental  Stars  and  the  one  published 
by  Dr.  DAVIS  in  his  Declinations  and  Proper  Motions 
of  Fifty-six  Stars.  The  places  of  the  stars  from 
the  Kam  Catalogue  were  brought  up  from  the  original 
places  published  in  the  Astronomisclie  Nachrichten. 


POSITIONS  OF  COMPARISON  STARS. 


Xo.  Epoch.  a 

1  1879.0  20h  12m  48s  .57 

2  "  20  7   3  .05 

3  1880.0  4  28  41  .13 

4  "  4  24  54  .27 

5  "  4  19  46  .85 


4  16  35  .23 

4  28   8  .06 
4  13   2  .70 


6  .44 


Wt. 
12 

ie 

5 

21 
£5 

30 

11 
51 


—  17°  51'  55".5 

-18  2  22  .2 

+  16  49  28  .2 

+  16  56  54  .6 

+  16  28  26  .0 

+  16  20  51  .2 

+ 16  56  58  .7 

+  16  ^13  54  .2 


15     37     55  .5 


Wt. 

12 

16 

5 

17 

27 

30 

11 
45 


17 


10  1882.0 

10 

25 

55 

.72 

20 

+  10 

1 

43 

.8 

20 

11 

10 

7 

8 

.49 

15 

+  H 

55 

20 

.5 

15 

12 

10 

7 

19 

.62 

18 

+  H 

56 

11 

.5 

18 

13 

10 

8 

55 

.30 

3 

+  12 

5 

41 

.3 

3 

14 

10 

12 

6 

.46 

21 

.+  11 

25 

37 

.4 

21 

15 


10       7       4  .97 


+  11     51     25  .5 


Authority. 

1/12   [4  Arg.   Weiss  16048  +  6  Kam.   3980  +  2  Mun.i  23872.] 
1/16  L'Mun.,  23483  +5  Mun.,  9920  +  10  Cape  (1890)  2511.] 
5  Konig.  Mer. 

1/21,  1/17    [W.  514  +  10  A.  G.  C.  Ber.  A.  1207  +  10,  6  Konig.  Mer.] 
1/25,  1/27  [W.  391  +  6,  8  Yarn.3  1952  +  10  A.  G.  C.  Ber.  A.  1178 

+  8  Konig,  Mer.] 

1/30  [Lai.  8214  +  W".  317  +4  Kam.  766  +4  Kam.  7(55  +  2,  5 
•      Yarn.3  1928  +  10  A.  G.  C.  Ber.  A.  1156  +  8,  5  Konig.  Mer.] 
1/11  [10  A.  G.  C.  Ber.  A.  1227  +  Eiim.  II,  2345.] 
1/51,  1/45  [Eiim.  II,  2345  +  3  Kam.  751  +  4  Kam.  752  +  10  Poul. 

1855,  634  +  6,  5  Yarn.3  1908  +  8,  3  Bad.  II,  480  +  10  Poul. 

1875,  980  +  10  A.   G.  C.  Ber.   A.   1138.] 
1/22,  1/17   [2  Kam.  719  +  10,  5  Yarn.3  1842  +  10  A.  G.  C.  Ber. 

A.  1102.] 

1/20  [5  Yarn..,  4476  +  5  Schj.  3856  +  10  A.  G.  C.  Leip.  I,  4048.] 
1/15   [W.  76  +  2  A.  N.  47,  135  +  10  Yarn.3  4341  +  2  Kam.   1696.] 
1/18  1 3  B.  B.  VI,  2169  +  6  Yarn.3  4344  +  3  Kam.  1697  +  6  A.  N. 

74,  247.] 

3  B.  B.  VI,  2178. 
1/21    [W.    173  +  5   Yarn.3   4380  +  5    Schj.    3772  +  10    A.   G.    C. 

Leip.  I,  3989.] 
1/6  [3  B.  B.  VI,  11°  2187  +  3  Kam.  1694.] 


OBSERVATIONS. 


The  following  observations  of  Psyche  were  collected  from  the  publications  mentioned  in  the  last  column 


Date. 

a 

5 

Place.         Inst. 

Star. 

Source. 

1879     Aug. 

4 

20h     5m 

3s 

.77 

-18° 

8' 

53". 

3 

Hamb.         M.  C. 

A.  N. 

Bd. 

97,   51. 

July 

29 

10 

0 

.00 

-17 

46 

34  , 

,2 

Krems. 

A.  N. 

Bd. 

97,  351. 

Aug. 

31 

2 

8 
6 

20 
39 

.08 
.63 

-18 

53 
1 

58  , 
38 

.1 
.4 

Green. 

Green. 

Obs 

.   1879. 

11 

19     59 

43 

.72 

33 

33 

.1 

"                  " 

" 

" 

" 

12 

59 

1 

.51 

36 

56  , 

,9 

«                  .< 

" 

" 

" 

July 

30 

20       9 

6 

.75 

-17 

50 

41 

.4 

Leipz.              E. 

1           A.  N. 

Bd. 

96,  225. 

Aug. 

1 

7 

27 

.45 

58 

7  , 

•5, 

"                  " 

2 

" 

"       " 

3 

5 

51 

.11 

-18 

5 

19 

.4 

2             " 

able  to  correct  SCHUBERT'S  elements  rather  than  to  have  used 
simply  a  block  of  five  oppositions  front  1879  to  1885.  Second — 
As  is  afterwards  shown,  it  is  not  possible  to  determine  empirical 
formulae  with  which  to  carry  forward  the  corrections  to  SCHU- 
BERT'S perturbations.  Third — 'The  perturbations  of  Psyche  by 
Saturn  are  of  considerable  magnitude,  and  these  should  have 


been  computed,  because  it  will  never  be  possible  to  get  a  satis- 
factory orbit  of  the  planet  by  neglecting  them.  In  the  light 
of  these  facts  it  is  deemed  unwise  to  try  to  carry  out  the 
second  part  of  the  program  outlined  above.  Instead  of  doing 
that,  it  is  the  author's  intention  to  compute  general  perturba- 
tions of  Psyche  by  both  Jupiter  and  Saturn. 


—  3  — 


1880 

Nov. 

10 

4  24 

14 

.()5 

4 

16 

40 

50 

.7      Hamb.    M.  C.        A.  N.  Bd.  100,  133. 

20 

20 

43 

.42 

31 

28 

.2        "        "          '   " 

Dec. 

23 

o 

18 

1 

.24 

24 
6 

37 

7 

.7 
.4 

3 

9 

0 

.06 

4 

13 

.9     Green.      "         Green.  Obs.  1880. 

11 

2 

25 

.29 

+ 

15 

52 

7 

.6 

Nov. 

27 

14 

21 

.92 

+ 

16 

15 

56 

.1      Paris               Comptes  Een.  Vol.  92. 

20 

20 

30 

.53 

30 

54 

.3      Wash.     T.  C.       Wash.  Obs.  Vol.  27. 

Dec. 

1 

10 

34 

.39 

7 

30 

.5 

6 

6 

15 

.20 

+ 

15 

58 

50 

2        "        "           "    " 

9 

4   3 

48 

.60 

54 

24 

.8 

13 

0 

46 

.52 

49 

37 

.5 

Nov. 

10 

29 

15 

.30 

+ 

16 

55 

25 

.6      Konig.     H.    3    A.  N.  Vol.  100,  243. 

15 

25 

12 

.62 

43 

30 

.5        "        "     4     "    "   " 

21 

24 

19 
17 

50 
6 

.77 
.80 

29 
22 

11 
23 

.3        "        "     5 

.4        "        "     6      "    "   " 

8 

30 

45 

.28 

59 

67 

.3      Leipz.      E.     7    A.  N.  Vol.  100,  357. 

29 

30 

12 
11 

33 

40 

.76 
.39 

11 

9 

51 
53 

.  -J                                                    O 

.7        "        "     8      "    "   " 

Dec. 

14 

4   0 

17 

.38 

+ 

15 

48 

56 

.6        "        "     9      "    "   " 

1882 

Feb. 

18 

10  26 

25 

.71 

+ 

9 

59 

57 

.6      Green.    M.  C.        Green.  Obs.  1882. 

Mar. 

9 

11 

52 

.15 

+ 

11 

37 

28 

.1      Paris       "         Comptes  Een.  Vol.  94. 

14 

8 

31 

.52 

59 

35 

.0        "        "           "     "    "   " 

15 

7 

53 

.90 

+ 

12 

3 

43 

.8        "        "           "     

16 
17 

7 
6 

17 

41 

.30 
.57 

7 
11 

45 
43 

.3        "        "           "     ' 

.7        "        "           "     "    "   " 

18 

6 

6 

.52 

15 

31 

.1 

20 

4 

59 

.90 

22 

55 

.2 

Feb. 

10 

09 

24 

85 

+ 

9 

18 

34 

.6     Wash.     T.  C.       Wash.  Obs.  1882. 

Mar. 

2 

16 

50 

.48 

+ 

11 

4 

30 

.6 

4 

15 

19 

.91 

14 

33 

.2 

14 

8 

23 

.31 

+ 

12 

0 

30 

.0 

Feb. 

18 

26 

31 

.54 

+ 

9 

59 

19 

.7      Wien      E.    10    A.  N.  Vol.  102,  283. 

Mar. 

13 

9 

13 

.77 

+ 

11 

54 

57 

.7        "        "    11 

14 

8 

32 

.50 

59 

28 

.6        "        "    12      "    "   "   " 

15 

7 

58 

.13 

+ 

12 

3 

15 

.1        "            13     "    ".  "   " 

7 

13 

20 

.36 

11 

27 

46 

.0      Dres.           14    A.  N.  Vol.  102,  187. 

8 

12 

34 

.35 

32 

49 

.8                    14     "    "   "   " 

12 

9 

54 

.54 

50 

27 

.3        "            15 

1883 

April 

13 

25 

10   9 
14  41 

16 
19 

.24 
.74 

- 

11 

54 
20 

38 
37 

.0                    15      "    "   "   " 
.0     Wash.    T.  C.        Wash.  Obs.  1883. 

27 

39 

46 

.07 

12 

24 

.1 

30 

37 

23 

.86 

0 

0 

.5 

May 

2 

35 

48 

.78 

— 

10 

51 

50 

.9 

3 

35 

1 

.51 

47 

50 

.7        "        "          "    "     " 

9 

30 

19 

.52 

24 

25 

.6 

2 

35 

59 

.25 

52 

45 

.4     Paris     M.  C.        Comptes  Ren.  Vol.  97. 

9 

30 

29 

.70 

"           "     "    "   " 

12 

28 

12 

.71 

" 

15 

26 

0 

.08 

3 

33 

.3        "        "           "     "    "   " 

16 

25 

16 

.87 

0 

11 

.6        "        "           "     "    "   " 

18 

14  23 

52 

.51 

— 

9 

53 

37 

.6        "        "           "     "    "   " 

1885 

Dec. 

1 

4  15 

6 

.50 

+ 

16 

19 

58 

.8                        Comptes  Een.  Vol.  102. 

11 
9 

12 

8 

26 
13 

.98 

.27 

14 
6 

28 
44 

.9 

23 

3  58 

37 

.09 

+ 

15 

55 

8 

.5 

30 

55 

28 

.71 

55 

52 

.2        "        "           "     "    "    " 

Arranging  these  observations  chronologically  and  have  been  determined  injtt^Mlt)«;i«§pwanner :  For 
comparing  them  with  the  ephemeris  values,  we  have  heliometer  and  microraetej',  .pb^ierva'tions  |he  weights 
the  following  table  of  residuals,  in  which  the  weights  (divided  by  six)  of  ttt§  compa^son  staife  (page  2) 


were  taken  as  the  weights  of  0  —  C.     When  clouds 
interfered  or  the'  seeing  was  bad  the  weights  were 
slightly  reduced.     Some  reduction  was  also  made  in 
the  weights  of  those  observations  dependent  upon  stars 
6  and  8,  because  on  account  of  the  large  number  of 
catalogue  places  of  these  stars  the  weights  became 
abnormally  large.     The  mean  of  the  weights  of  those 
observations   dependent   upon   A.    G.    C.   stars   is   4. 
This  weight,  4,  was  given  to  each  of  the  meridian  and 
transit   circle   observations.     All  weights   were   then 
divided  by  four  and  these  Avere  used  in  taking  the 
weighted  means.    The  residuals  in  declination  for  the 
Kremsmunster  observations  of  1879  July  29  and  31 
differ  about  10"  from  the  residuals  of  the  observations 
made  at  Hamburg,  Paris,  and  Leipzig.     Perhaps  an 
error  of  10"  was  made  in  reading  the  Kremsmunster 
circle,  but  no  further  evidence  to  support  such  a  sup- 
position could  be  found.     It   is  my  judgment  that 
more  might  be  lost  by  including  these  residuals  than 
by  rejecting  them  ;  hence  zero  wreight  has  been  as- 
signed to  each. 

RESIDUALS. 
Date.            0  —  C         W            O  —  C         W      Remarks. 
1879. 
July    29          —4s  .67         0.75         4    1".7         0.0 
3Q          —  4  .73           .5              -    9  .2           .5         Clouds. 
31          -4  .72           .75         4'    2  .1           .0 
Aug.     1          -4  .94           .5            -13  .1           .5         Clouds. 
2          -4  .98         1.                -    9  .5         1. 
3         —5  .00           .5            -11  .9           .5         Clouds. 
4         -5  .01         1.              -11  .6         1. 
11          -5  .13         1.                -    9  .0         1. 
12          —5  .07         1.              -10  .5         1. 

1882. 
Feb.  10         44S.47         1.00          —  24".8         1.00 
18              4  .60           .75              20  .4           .75 
18              4  .32         1.                  20  .4         1. 
Mar.     2              4  .53         1.                  24  .3         1. 
4              4  .53         1.                  24  .0         1. 
7              4  .48         1.                  23  .3         1. 
8              4  .24         1.                  22  .3         1. 
9              4  .52         1.                  24  .8         1. 
12              4  .18           .25              19  .7           .25 
13              4  .05           .25              20  .9           .25 
13              4  .56           .5                20  .9           .5 
14              4  .38         1.                  20  .5         3. 
14              4  .42           .75              20  .8           .75 
14              4  .54         1.                  20  .7         1. 
15              4  .06           .125            18  .7           .125 
15              4  .37         L                  19  .9         1. 
16              4  .47         1.                  20  .7         1. 
17              4  .53         1.                   18  .7         1. 
'  18              4  .27         1.                  21  .5         1. 
Mar.  20         44  .30         1.              -18  .5         1. 

Mar.     9         4  4s  .425     16.625*     —  21".55     16.625 

1883. 
Apr.  25         4  Is  .23         1.              —  3".4         1. 
27              1  .33         1.                  6  .9         1. 
30              1  .13         1.                  4  .1         1. 
May     2              1   .12         1.                  3  .4         1. 
2              1  .11         1.                  4  .3         1. 
3              1  .36         1.                  5  .5         1. 
9              1  .17         1.                  8  .1         1. 
9              1  .16         1. 
12              1  .14         1. 
15              1  .17         1.                  4  .4         1. 
16              1   .06         1.                  5  .2         1. 
May  18         4  1   .00         1.              —2  .5         1. 

Aug.     3          -4s  .938       7.0            -  10".48       5.5 

1880. 
Nov.     8         -)-  5s.  28         0.5           4    8".0         0.5 
10              5  .30           .25                8  .4           .25 
15              5  .52           .75                5  .7           .75 
16              5  .26         1.                    6  .5         1. 
20              5  .34         1.                    8  .9         1. 
20              5  .43         1.                    8  .5         1. 
21              5  .33         1.                     7  .2         1. 
23              5  .50         1.                    8  .8         1. 
24              5  .39         1.                    7  .5         1. 
27              5  .43         1.                  10  .8         1. 
29              5  .32         1.25              10  .8         1.25 
30              5  .31         1.25              10  .8         1.25 
Dec.      1              5  .47         1.                  10  .2         1. 
2                                                    11  .8           .5      Clouds. 
3              5  .34         1.                   10  .2         1. 
6              5  .18         1.                  10  .8         1. 
9              5  .12         1.                  10  .0         1. 
11              5  .13         1.                  11  .1         1. 
13              5  .04         1.                  11  .5         1. 
Dec.    14        4.4  .92         0.75         4  11  .5         0.75    Bad  seeing. 

May      7         4  Is  .165     12.              —  4".78     10. 

1885. 
Dec.      1         44s.63         1.             4  17".9         1. 
4              4  .56         1.                  10  .3         1. 
9              4  .66         1.                  13  .0         1. 
23              4  .34         1.                  11  .9         1. 
Dec.   30         44  .33         1.             4  11  .2         1. 

Dec.     13         4  4s  .504       5.             4  12".86       5. 

No  smoothing   out   process  has   been   applied  to 
the  residuals,  and  on  account  of  the  small  number  of 
observations  in  most  of  the  series  the  computation 
of  probable  errors  for  the  purpose  of  weighting  the 
residuals  has  been  avoided.     It  was  thought  that  the 
method  of  procedure  adopted  would  give  just  as  reli- 
able results  as  a  more  elaborate  process. 

Nov.   27         45s.298     17.75         4    9".53     38.25 

5 

DIFFERENTIAL  COEFFICIENTS  AND  LEAST  SQUARE  SOLUTION. 

Transforming  the  ephemeris  positions  of  the  planet  for  the  dates  of  the  normals  to  the  epoch  of  1870, 
we  have : — 


a 

5 

log  A 

1879 

August 

5.5 

300° 

56' 

16" 

.5 

-18° 

14' 

5" 

.8 

0.240293 

1880 

November 

29.5 

62 

57 

27  , 

,3 

+  16 

9 

58  , 

.3 

0.218635 

1882. 

March 

7.5 

153 

8 

4  . 

,1 

+  11 

32 

19  , 

o 

0.352240 

1883 

May 

7.5 

217 

49 

26  , 

2 

—  10 

29 

16 

.7 

0.354937 

1885     December 


13.5 


61 


1     11  .7 


4-  15     59       6  .4 


0.234298 


AVith  these  coordinates  and  SCHUBERT'S  elements,  differential  coefficients  were  computed  giving  the  follow- 
ing logarithmic  observation  equations : — 


OBSERVATION  EQUATIONS. 


0.189186  ATT  -f  9.155347n  V^Att  -\-  9.044763n  At  -f  0.485258n 


0.237386 


0.194755 

4-  7.938927n 

4-  9.467439 

4-  0.409537 

4-  0.280214 

4-  0.880449 

0.132406 

4-  9.400944n 

4-  8.614869 

4-  0.222787 

4-  0.047791 

4-  0.694169 

0.138764 

4-  8.976777n 

4-  9.606266 

4-  0.068281n 

4-  0.036684 

4-  0.723934 

0.179010 

4-  7.532970 

4-  9.475303 

4-  0.399199 

4-  0.261723 

4-  1.025690 

9.400352 

4-  9.870979 

4-  9.835881 

4-  9.691681U 

4-  9.451835 

4-  9.995812 

9.467262 

4-  8.892532n 

4-  0.192931n 

4-  9.649330 

4-  9.560284 

4-0.157535 

9.636191n 

4-  9.857489n 

4-  9.110204 

4-  9.732451n 

4-  9.552548n 

4-  0.198388n 

9.629199n 

4-  9.371875n 

4-  0.114306 

4-  9.524342 

4-  9.524494n 

4-  0.209925n 

9.469337 

4-  9.115697n 

4-  0.176690n 

4-  9.664063 

4-  9.558994 

4-  0.321169 

The  weights  assigned  are  very  approximately  those  of  the  weighted  residuals.     Each  equation  was  multi- 
plied through  by  the  square  root  of  its  weight,  divided  by  100,  and  the  following  substitutions  were  then  made : — 


[8.345270]  ATT     =  x 

[8.008004]   1/10  Afl    =  y 
[8.343446]  A«     =  z 


[8.560052]  A</>    =  u 

[8.430729]  A.¥0  ==  v 

[9.030964]   1000  A/*     =  w 


The  resulting  weighted  homogeneous  observation  equations  are  as  follows : — 


9.843916  x  4-  9.147343n 

y  4-  8.701317n 

s  -\-  9.925206n 

u  4-  9.806657  r  4-  9.749106  «•  4-  9.867266 

=  0 

0.000000 

4-  S.081438n 

4-  9.274508 

4-  o.oooooo 

4-  0.000000 

4-  o.oooooo 

4-  0.033196n 

=  0 

9.937651 

4-  9.543455n 

4-  8.421938 

4-  9.813250 

4-  9.767577 

4-  9.813720 

4-  9.963652n 

=  0 

9.881539 

4-  9.056818n 

4-  9.350865 

4-  9.596274n 

4-  9.694000 

4-  9.781015 

4-  9.323144n 

=  0 

9.833740 

4-  7.524966 

4-  9.131857 

4-  9.839147 

4-  9.830994 

4-  9.994726 

4-  9.812562U 

—  0 

9.055082 

4-  9.862975 

4-  9.492435 

-f  9.131629a 

4-  9.021106 

4-  8.964848 

+  9.020361 

=  0 

9.272507 

4-  9.035043n 

4-  0.000000n 

4-  9.239793 

4-  9.280070 

4-  9.277086 

4-9.129608n 

=  0 

9.441436n 

4-  0.000000n 

4-  8.917273 

4-  9.322914n 

4-  9.272334n 

4-  9.317939U 

4-  9.483962 

=  0 

9.371974n 

4-9.451916n 

4-  9.858905 

4-  9.052335 

4-  9.181810n 

4-  9.267006n 

4-  8.767473 

=  0 

9.124067 

4-  9.107693n 

4-  9.833244n 

4-  9.104011 

4-  9.128265 

4-  9.290196 

4-9.109241n 

=  0 

Using  the  method  of  least  squares,  the  normal  equations  are  (numerical  coefficients)  :— 
4-3.4798  x  —  0.1090  y  -\-  0.0036  z  4-  1.2117   w  4-  2.9460  f  4- 3.2623  w  —  2.0934  —  0 


—  0.1090 

4-  1.7953 

4-  0.1056 

—  0.0295 

—  0.0919 

—  0.1019 

4-  0.0389  =  0 

4-  0.0036 

4-  0.1056 

4-  2.1969 

4-  0.0145 

—  0.0006 

4-  0.0022 

—  0.0771  =  0 

4-  1.2117 

—  0.0295 

4-  0.0145 

4-  2.8858 

4-1.1727 

4-  1.4629 

—  2.7746  =  0 

4-  2.9460 

—  0.0919 

—  0.0006 

4-  1.1727 

4-  2.5804 

4-  2.8479 

-1.7881  =  0 

4-  3.2623 

—  0.1019 

4-  0.0022 

4-  1.4629 

4-  2.8479 

4-  3.2396 

—  2.1488  =i  0 

The  similarity  of  the  coefficients  in  the  first  ar^d 
sixth  of  these  equations  shows  that  the  value  of  the 
unknowns  determined  from  them  will  be  somewhat 


uncertain.  The  "process  of  solution  Avas  carried  out, 
however,  and  the  following  elimination  equations 
derived : — 


4-  3.4798  x  —  0.1090  y  -f  0.0036  s  -f  1.2117  u  -f  2.9460  v  -\-  3.2623  w  —  2.0934  —  0 
4.  1.7919 


4-  0.1057 

+  0.0084 

4-  0.0004 

4-  0.0003 

—  0.0267  —  0 

4-  2.1907 

4-  0.0127 

—  0.0036 

-  0.0012 

—  0.0733  =  0 

4-  2.4638 

-j-  0.1469 

4-  0.3270 

—  2.0452  —  0 

4-  0.0775 

4-  0.0665 

4-0.1060  =  0 

4-  0.0808 

—  0.0059  =  0 

whence 


w  =  4-  0.0730 
v  —  —  1.4303 
u  =  0.7351 


The  usual  check  quantities  were  carried  through- 
out the  formation  and  solution  of  the  normal  equa- 
tions. The  smallness  of  the  coefficients  in  the  last 
two  elimination  equations  makes  the  determination  of 
v  and  w  uncertain  ;  and  the  uncertainty  of  these  also 
affects  the  determination  of  the  other  quantities, 
especially  x. 

In  order  to  determine  the  unknowns  more  accu- 
rately the  first  four  of  the  elimination  equations  were 
chosen,  and  by  successive  substitutions  u,  z,  y,  and  x 
were  expressed  as  functions  of  v  and  w,  giving  the 
following  equations  :  — 


u  —  8.77542n  v  4-  9.12295n  w  4-  9.91913 

s  —  7.29863   +  7.11965  +  8.45708 

y  —  5.78568n   4-  6.57640  4-  7.96937 

x  —  9.91690n   4-  9.95001n  4-  9.49528 


(A) 


Substituting  these  values  of  u,  z,  y,  and  x  in  the 
original  homogeneous  weighted  observation  equations, 
we  have  the  following  set  of  equations  for  the  deter- 
mination of  v  and  w: — 


9.05806  v  4-  8.70422  w 
9.06032   4-  8.37561n 
9.22671n  4-9.31658n 
9.04297n  4-8.34031n 
8.86656   4-  9.46044 
8.29842   4-  7.98687 
8.37234   4-  7.31511n 
8.73056   4-  8.81948 
8.57060   4-  8.03448 
8.19166   4-  8.76841 
Check  [m'  m']  =  0.2021, 

4-  9.40401  =  0 
4-  8.83731  =  0 
4-  9.04634n  =  0 
4-  9.46924n  —  0 
4-  9.14898  =  0 
4-  8.63960  —  0 
4-  8.58364  =  0 
4-  8.56549  =  0 
4-  8.98490  =  0 
4-  7.34452n  —  0 
[m  m  4]  =  0.2019 

These  equations  were  multiplied  through  by  the 
number  whose  logarithm  is  0.53076  and  new  un- 
knowns defined  by  the  relations  v'  =  [9.75747]  v, 
w'  =  [9.99120]  w  were  introduced.  Solution  by  the 
method  of  least  squares  gave  for  the  normals, 

4-  2.7365v'  +  1.3698-w;'  +  2.1313  =  0 
4-  1.3698v'  4-  1.6550w'  +  1.0004  =  0 


3  =  -\-  0.0269 
y  =  +  0.0102 
x  =  -\-  1.4884 


from  which 


and  hence 


v'  —  —0.8132 
w'  —  +0.0686 
v  =  —1.4214 
w  =  +0.0700 

These  values  of  v'  and  w'  substituted  in  the  obser- 
vation equations  from  which  they  were  determined 
give  [v'  v'}  =  0.6639.  This  divided  by  the  square  of 
the  number  whose  logarithm  is  0.53076  gives  [v  v]  =: 
0.0576,  while  [m  m.6]  —  0.0565,  which  is,  I  think, 
considering  the  inherent  uncertainty  of  the  solution, 
a  satisfactory  agreement. 

Substituting  the  values  of  v  and  w  in  equations  A, 
we  have  finally, +  1  4243 

+  0.0094 
+  0.0259 
4-  0.9056 
- 1.4214 
4-  0.0700 


x 

y 

z 
u 

V 

w 


Returning  now  to  the  relations  between  these  quan- 
tities and  the  original  unknowns   (page  5)  we  have 
finally  the  following  corrections  to  the  elements  :— 
A7r  —  4-64".  32 
Aft  —  4-    9  .26 
Ai  —  +    1  .18 
A<£  —  4.  24  .94 
AM0  =  —  52  .72 
A^  —  +    0  .000652 

Substituting  these  values  in  the  original  observa- 
tion equations,  we  have  [pvv]  =  576".32,  which  divid- 
ed by  10,000  gives  an  exact  agreement  with  [vv]  just 
found.  On  account  of  the  small  number  of  observa- 
tion equations  the  computation  of  probable  errors  of 
corrections  to  the  elements  has  been  carefully  avoided. 
As  a  check  upon  the  work  an  ephemeris  was  computed, 
with  the  corrected  elements,  for  each  of  the  normal 
dates  giving  the  following  coordinates,  referred  to  the 
ecliptic  and  equinox  of  1870.0 : — 


1879  August  5.5 

1880  November  29.5 

1882  March  7.5 

1883  May  7.5 
1885  December  13.5 

A  comparison  of  these  coordinates  with  the  observed 
coordinates  gives  the  following  residuals,  those  in  a 
having  been  multiplied  by  cos  8 : — 


300°  55'  10".6 
62  58  36  .7 
153   9  16  .0 
217  49  41  .2 
61   2  29  .6 

—  18°  14'  14".6 
+  16  10   7  .6 
+  11  31  55  .3 
—  10  29  20  .6 
+  15  59  18  .0 

O  —  C 

0  —  C 

in  a 

in  8 

-7".  8 

—  1".9 

+  6  .2 

-0  .3 

-7  .5 

+  2  .7 

+  11  .2 

-3  .9 

—  5  .3 

+  2  .4 

From  these  we  find  [pvv]  =  513".3,  which,  consid- 
ering the  element  of  indeterminateness  inherent  in 
the  least  square  solution,  is  perhaps  in  satisfactory 
agreement  with  the  value  576".3  found  from  the  least 
square  solution.  The  value  of  [pvv]  before  the  least 
square  solution  was  31,003". 7.  The  above  residuals 
are  still  rather  large,  but  if  we  consider  the  fact,  as 
is  afterwards  shown,  that  SCHUBERT'S  perturbations 
are  probably  not  strictly  accurate,  and  also  that 
perturbations  by  Saturn  and  other  planets  have  not 
been  computed,  it  seems  useless  to  try  and  further 
reduce  these  residuals. 

The  corrected  elements  are, 
Epoch  and  osculation  1870  Jan.  0.  B.  M.  T. 


On  account  of  the  uncertainties  of  the  least  square 
solution,  it  is  certain  that  these  elements  are  not 
exact  to-ihe  tenths  of  a  second,  but  these  have  been 
retained  because  it  is  customary  to  print  the  elements 
in  that  way  in  the  Jahrbuch. 

Special  Perturbation*.  From  the  smallness  of  the 
corrections  to  SCHUBERT'S  elements  it  is  evident  that 
practically  the  same  perturbations  should  be  obtained, 
no  matter  which  set  of  elements  is  used  in  the  com- 
putation. Nevertheless,  in  order  to  be  perfectly 
sure  about  the  matter,  special  perturbations  by  Jupiter 
were  computed,  with  the  corrected  elements,  from 
1870  to  1876.  These  perturbations,  however,  showed 
much  greater  differences  from  SCHUBERT'S  than  was 
to  be  expected,  amounting  to  60"  in  dw  and  dM. 

The  method  of  variation  of  constants  was  used, 
( formulas  from  WATSON,)  and  the  perturbations  were 
computed  for  forty-day  intervals  upon  the  same  dates 
as  were  used  by  SCHUBERT.  To  avoid  the  accumula- 
tion of  perturbations  of  the  second  order,  the  elements 
were  corrected  for  the  perturbations  at  frequent  in- 
tervals, the  first  date  of  a  new  computation  being 
always  the  same  as  the  last  date  of  the  preceding  one. 
When  Psyche  was  nearest  Jupiter  this  interval  was 
about  six  months,  but  when  furthest  from  Jupiter  it 
could  be  taken  as  great  as  one  and  one-half  years. 
BESSEL  's  mass  of  Jupiter  was  used  and  i'  and  O'  were 
taken  from  the  American  Ephemeris,  for  these  are 
probably  the  values  used  by  SCHUBERT,  as  he  was  at 


M 

330° 

59' 

12".  8 

that  time  a  computer  in  the  Nautical  Almanac  Office. 

7T 

15 

51 

33 

.7  \  Ecliptic  and 

The  longitude  and  the  radius  vector  of  Jupiter  were 

O 

150 

35 

32 

.9  i  mean  equinox 

taken  from  SCHUBERT'S  tables 

published  in  A.  N., 

i 

3 

3 

59 

.9)  of 

1870.0. 

vol.  75,  p. 

209. 

^ 

0 

7 

49 

21 

.2 

!"• 

710" 

.7200 

The  following  table, 

taken  in 

the  sense  TOWNLEY  — 

log  /* 

2 

.851699 

SCHUBERT, 

exhibits 

the    differences     in    the    per- 

log   a 

0. 

46554 

turbations 

:  — 

Berlin  M.  T. 

Adi 

Adfi 

Ad(/ 

>            Ad?r 

AdM 

A/<fc 

MM 

1870 

October 

7.0 

0" 

0" 

0 

—    3" 

+  0".002 

0" 

+    4" 

November 

16.0 

0 

—  1 

0 

—    1 

+  0  .000 

+    4 

+    2 

1870 

December 

26.0 

0 

1 

+  1 

—    4 

+  0  .009 

0 

+    5 

1872 

January 

30.0 

0 

—  1 

+  2 

+  10 

+  0  .003 

+  1 

—  12 

March 

10.0 

0 

1 

+  2 

+  13 

+  0  .003 

+  1 

—  15 

1872 

April 

19.0 

0 

—  ! 

+  2 

+  15 

+  0  .004 

+  1 

—  18 

1873 

March 

5.0 

o, 

—  0 

+  4 

+  25 

+  0  .012 

+  3 

—  28 

April 

14.0 

0 

—  1 

+  5 

+  26 

+  0  .013 

+  4 

-29 

1873 

May 

24.0 

0 

-1 

+  5 

+  27 

+  0  .014 

+  5 

-31 

1874 

June 

28.0 

0 

—  0 

+  6 

+  44 

+  0  .019 

+  10 

—  46 

August 

7.0 

0 

1 

+  6 

+  46 

+  0  .019 

+  12 

—  47 

1874 

September 

16.0 

0 

—  1 

+  6 

+  47 

+  0  .019 

+  12 

—  48 

1875 

October 

21.0 

0 

-1 

+  7 

+  58 

+  0  .021 

+  20 

—  59 

November 

30.0 

0 

-1 

+  7 

+  59 

+  0  .020 

+  20 

—  60 

1876 

January 

9.0 

0 

—  1 

+  7 

+  60 

+  0  .020 

+  21 

—  60 

Discussion  of  Differences  in  Perturbations.  Al- 
though it  seemed  practically  certain  that  these  differ- 
ences between  the  two  sets  of  perturbations  were  not 
due  to  the  differences  in  the  elements  used,  yet  it  was 
deemed  advisable  to  investigate  the  point  further. 
It  was  thought  that  perhaps  a  few  of  these  differences 
in  the  elements  had  a  controlling  influence  in  produc- 
ing the  differences  in  the  perturbations,  and  if  such 
were  the  case,  it  would  be  possible  to  construct  em- 
pirical formulae  by  which  to  represent  the  effect,  of 
these  increments  to  the  elements.  On  account  of  its 
simplicity  the  formula  for  the  computation  of  the 
perturbation  in  fl  was  investigated  first.  This  for- 


mill  a  is  —jj-j-  : 


r  sin  u    ., 
sin  i 


Let  us  for  brevity  put 


=  A. 

dt 


The  values  of 


A  were  computed  to  hundredths  of  a  second  of  arc, 
and  any  slim  of  changes  in  the  factors  of  the  right- 
hand  side  of  the  equation  above,  which  produce  a 
change  of  less  than  0".005,  can  be  considered  as  neg- 
ligible. A  change  in  the  logarithm  sufficient  to  pro- 
duce a  change  of  0".00o  in  A  would,  on  the  average, 
produce  a  change  of  0".01  in  A  in  half  the  cases,  if 
the  values  of  A  are  taken  to  hundredths.  A  change 
in  the  logarithm  sufficient  to  produce  a  change  of 
0".004  in  A  would,  in  four  cases  out  of  ten,  produce  a 
change  of  0".01  in  A,  but,  likewise,  a  change  in  the 
logarithm  sufficient  to  produce  a  change  of  0".006  in 
A  would  not,  four  times  out  of  ten,  produce  a  change 
of  0".01  in  A.  Likewise  for  0".003  and  0".007,  etc. 

During  the  six  years  over  which  perturbations 
were  computed  the  maximum  value  of  A  was  53", 
reached  at  perijovian  distance,  1871  April  5.  During 
most  of  the  time,  however,  its  value  was  much  less 
than  this.  The  changes  in  the  logarithm  of  A  neces- 
sary to  produce  a  change  of  0".005  in  A  are  exhibited 
in  the  following  table: — 


A 
53" 

log  A 
1.72428 

A  log  A 
0.00004 

50 

1.69897 

4 

45 

1.65321 

5 

40 

1.60206 

5 

35 

1.54407 

6 

30 

1.47712 

7 

25 

1.39794 

9 

20 

1.30103 

11 

15 

1.17609 

15 

10 

1.00000  • 

0.00022 

During  the  time  over  which  perturbations  were 
computed  by  me  the  value  of  A  exceeds  15"  on  only 


ten  of  the  forty-day-interval  dates.  By  neglecting, 
then,  any  changes  in  log  A  of  15  units  or  more  in  the 
fifth  place,  an  error  of  at  least  0".005  will  be  intro- 
duced into  each  of  these  values  of  A.  If  we  assume 
that  a  change  of  15  units  in  the  fifth  place  has  been 
made,  then  the  effect  on  the  ten  values  of  A  above  15" 
will  be  as  exhibited  in  the  following  table: — 


A 

near. 

No. 

Error. 

20" 

2 

0".014 

30 

2 

0  .020 

40 

2 

0  .026 

50 

4 

0  .068 

Total, 


0".128 


As  Psyche  came  into  perijovian  distance  four  times 
during  the  period  under  consideration,  1870  to  1900, 
then,  assuming  that  the  effect  will  be  the  same  each 
time,— -that  is,  neglecting  eccentricities, — we  have  for 
the  total  effect  upon  dft,  4  X  0".128  =  0".512.  As 
SCHUBERT'S  perturbations  are  published  to  seconds, 
this  would  a  little  overreach  the  limit  of  error  allow- 
able. It  appears,  then,  that  we  may  safely  neglect  any 
difference  that  will  not  produce  a  change  of  10  units 
in  the  fifth  place  of  logarithms,  provided  we  guard 
against  the  possible  piling  up  of  several  changes  of 
smaller  magnitude. 

With  these  preliminary  investigations,  we  may 
now  take  up  the  investigation  of  the  individual  factors 
in  the  equation  for  A.  k  has  the  same  value  in  both 
computations,  but  \/p,  sin  i,  r,  sin  u  and  Z  have 
slightly  different  values.  The  increments  are  all 
small,  as  the  following  analysis  will  show :  \/p  — 
V»  cos  (f>;  a  is  a  function  of  p,  and  as  we  have  the 
increment  of  /*,  A/t  =  -j-  0.00065  (see  page  6),  the 
increments  of  \/a  and  \/p  may  be  computed  by  dif- 
ferentiation, as  follows: — 


=  A  V<*  cos  <£  —  \/a  sin 


Carrying   out   the   computations   according   to   these 
formula1,  we  find, 


lo 


lo 


=  0.23277  log  \/p  =  0.22871 

=  3  Jl820n  —  10,  log  A  \/p  =  5.45806U  —  10 


And  from  a  subtraction  logarithm  table  it  is  easily 
seen  that  this  value  of  log  A\/P  will  produce  a  change 
of  only  seven  units  in  the  sixth  place  of  log  \/P- 
The  increment  of  r  is  small,  but  not  constant,  on 


account  of  the  increased  eccentricity  of  the  ellipse 
and  of  the  changing  value  of  Av.  It  is  my  purpose 
to  consider  at  present  only  the  former  of  these.  Ar 
will  then  have  a  maximum  value  when  the  planet  is 
at  an  extremity  of  the  minor  axis,  and  a  minimum 
value  at  an  extremity  of  the  major  axis.  At  this 


.]/  =  M0  +  tp. 
M=  2e  sin  M 


2M 


sin  3M  - 


latter  point  Ar  =  Art  =  - 


i,  which,  how- 


ever,  is  not  sufficient  to  produce  any  change  in  the 
sixth  place  of  log  a.  The  change  in  r  would  be 
greater  than  this  at  perihelion  and  less  at  aphelion. 
Since  &—  a  cos  </>,  the  increment  of  b  due  to  the  in- 
crements of  a  and  <£,  may  be  computed  from  the  rela- 
tion A&  =  Act  cos  <f>  —  a  sin  </>:  A</>  sin  I".  The  result 
is  log  A&  =  5.6819n,  which  would  produce  a  change 
of  eight  in  the  sixth  place  of  log  &. 

At  rr  +  1".2,  which  makes  A  log  sin  i,  47  units 
in  the  sixth  place. 

The  difference  sin  UT  — sin  «s,  in  which  the  sub- 
scripts stand  forTowNLEY  and  SCHUBERT,  can  be  found 
from  the  relation  u  =  v  -j-  TT  —  O,  provided  Av  can 
be  found.  This  can  be  done  from  the  known  rela- 
tion between  v  and  M,  AM0  being  already  known.  We 
have : — 


3sinM)  +  ...... 

hence  differentiating 

AM  =  AM0  -f  /A/* 

Ai>  =  AM  j  1  -+-  2c  cos  M  -f  5/2e2  cos  2M  j  +  Ae 

(  e2  } 

-.  e  sin  M+5/2e  sin  2M-f  ~j(13  sin  3M—  3  sinM)-f  .  .  Y 

Omitting  for  the  present  the  term  £A/*,  and  calling 
the  corresponding  values  of  Ai>  by  Av0,  we  have, 
taking 

e  =  0.136,  e-  =  0.0185,  r'/2e2  =  0.0462,  AM0  =  —  52".  7, 

Ae  =  +  24".7 

M  0°  45°  90°  135°  180°  225°  270°  315° 
A^0  —70"  —18"  —3"  —15"  —41"  —70"  —98"  —107" 
Since  ATT  —  AC  =  +  55",  the  values  of  Aw  will  lie 
between  -f-  52"  and  —  52".  A  log  sin  u  will  vary 
greatly,  depending  upon  the  value  of  u.  The  period 
of  u  is  five  years,  the  sidereal  period  of  Psyche.  The 
effect  of  A?t  is  indicated  in  the  following  tabulation. 
The  small  term  /A/A  has  been  allowed  for  in  the  values 
of  Av. 


Date. 

M 

u 

Av 

AM 

A  log  sin  u 

A 

AA 

1869 

Dec. 

11 

327° 

182° 

28' 

-  107" 

—  52' 

-0.00254 

+  0" 

.01 

—  0" 

.0000 

1870 

Apr. 

10 

351 

212 

57 

-  84 

—  29 

10 

2 

.34 

+  0  . 

0006 

Aug. 

8 

34 

244 

20 

—  48 

+  7 

+      1 

—  13 

.96 

—  0  . 

0003 

Dec. 

6 

38 

274 

27 

—  23 

+  32 

1 

—  39 

.03 

+  0  . 

0009 

1871 

Apr. 

5 

62 

301 

52 

-  10 

+  45 

.     Q 

—  53 

.12 

+  0  . 

0075 

Aug. 

3 

85 

326 

23 

—  3 

+  52 

17 

—  30 

.53 

+  0  . 

0121 

Dec. 

1 

109 

348 

19 

—   7 

+  48 

49 

—  6 

.73 

+  0  . 

0077 

1872 

Mar. 

30 

133 

8 

18 

-  13 

+  42 

+     60 

+  2 

.49 

.  +0  . 

0034 

July 

28 

157 

27 

5 

—  27 

+  28 

+     12 

+  3 

.69 

+  0. 

0010 

Nov. 

25 

180 

45 

15 

•  41 

+  14 

+      3 

+  2 

.33 

+  0  . 

0002 

1873 

Mar. 

25 

204 

63 

27 

•  O  / 

—  2 

0 

+  o 

.80 

0 

0000 

July 

23 

228- 

82 

15 

—  72 

-17 

1 

—  0 

.07 

+  0  . 

0000 

Nov. 

20 

252 

102 

18 

-  89 

-34 

+      1 

—  0 

.21 

—  0  .0000 

1874 

Mar. 

20 

275 

124 

18 

-  99 

—  44 

+      6 

+  o 

.08 

+  0  . 

0000 

July 

18 

299 

148 

54 

-105 

—  50 

+     18 

+  o 

.32 

+  0  . 

0001 

Nov. 

15 

323 

176 

33 

-107 

—  52 

+  0.00181 

+  ° 

.07 

+  0  . 

0003 

Taking  up  now  the  investigation  of  Z,  we  have : 
Z  =  m'  k2  h  r'  sin  ft.  No  statement  is  made  by 
SCHUBERT  concerning  the  value  of  the  mass  of  Jupiter, 
or  of  k  used  by  him.  For  the  present  I  shall  assume 
that  we  have  used  the  same  values.  The  increment  of 
h  depends  upon  that  of  p,  which  latter  might  be  found 
by  differentiating  the  expression  p-  =  r'2  +  r2  — 
2rr'  cos  ft  cos  (w1  —  w).  It  will,  however,  be  suf- 


ficient for  our  present  purpose  to  use  approximate  val- 
1  1 


ues  of  A/o.  h  =1 


,  hence  A/i  —  — 3p'4  Ap  = 


—  3p-*  Ar  for  a  maximum  value.  Taking  now  Ar  = 
V2  (Aa  +  A&),  a  mean  value,  neglecting  the  incre- 
ment of  v,  we  have  the  effect  of  A/i  illustrated  in  the 
following  table: — 


—  10  — 


Date. 

log  VP4     log  A7i 

A  log  li 

A 

AA 

1869 

Dec. 

11 

7.1913 

3 

.0493 

0.00024 

-f  0" 

.01 

0".0000 

1870 

Apr. 

10 

7.5154 

3 

.3733 

2 

,  2 

.34 

1 

Aug. 

8 

7.9098 

3 

.7678 

1 

-13 

.96 

3 

Dee. 

6 

8.2984 

4 

,1564 

1 

—  39 

.03 

9 

1871 

Apr. 

5 

8.5052 

4 

,3632 

2 

—  53 

.12 

25 

Aug. 

3 

8.4223 

4 

.2803 

2 

—  30 

.53 

14 

Dec. 

1 

8.1709 

4 

.0289 

1 

—  6 

.73 

2 

1872. 

Mar. 

30 

7.8931 

3 

.7511 

1 

i   2 

.49 

1 

July 

28 

7.6397 

3.4977 

1 

+  3 

.69 

1 

Nov. 

25 

7.4182 

3 

.2762 

2 

i   2 

.33 

1 

1873 

Mar. 

25 

7.2224 

3 

.0804 

3 

+  ° 

.80 

1 

July 

23 

7.0468 

2 

.9048 

15 

-  0 

.07 

0 

Nov. 

20 

6.8870 

2 

.7450 

2 

—  0 

.21 

0 

1874 

Mar. 

20 

6.7420 

9 

.6000 

0 

+  ° 

.08 

0 

July 

18 

6.6139 

2 

.4719 

0 

+  ° 

.32 

0 

/ 

Nov. 

15 

6.5083 

2 

.3663 

0.00000 

+  ° 

.07 

0  .0000 

The  signs  of  these  corrections  have  not  been  con- 
sidered. 

ft'  is  the  heliocentric  latitude  of  the  disturbing 
planet  with  respect  to  the  fundamental  plane — the 
instantaneous  orbit  of  the  disturbed  planet  in  this 
case.  The  maximum  possible  value  of  A/3'  is  therefore 
A*.  The  values  of  A  sin  ft'  could  be  computed  from 
its  known  relations,  but  so  many  auxiliary  quantities 
enter  that  the  process  might  become  quite  compli- 
cated. The  maximum  value  of  ft'  is  2°  28'.  If  A/2' 
have  its  maximum  value  when  ft'  is  a  maximum,  then 
A  log  sin  ft'  would  amount  to  six  units  in  the  fifth 
place  of  decimals.  The  following  tabulation  shows 
the  effect  of  a  possible  maximum  A/3'  during  the 
interval  under  consideration: — 


Max.  A 

Date.           /3' 

log  sin  ft' 

A 

AA 

1869 

Dec. 

11 

2° 

0' 

0.00007 

+  0" 

.01 

O^'.OOOO 

1870 

Apr. 

10 

2 

14 

6 

—  2 

.34 

3 

Aug. 

8 

2 

23 

6 

-13 

.96 

19 

Dec. 

6 

2 

28 

6 

—  39 

.03 

55 

1871 

Apr. 

5 

2 

27 

6 

—  53 

.12 

75 

Aug. 

3 

2 

22 

6 

—  30 

.53 

43 

Dec. 

1 

2 

13 

6 

-  6 

.73 

10 

1872 

Mar. 

30 

2 

0 

7 

+  2 

.49 

4 

July 

28 

1 

44 

8 

+  3 

.69 

7 

Nov. 

25 

1 

25 

10 

+  2 

.33 

6 

1873 

Mar. 

25 

1 

4 

13 

-f  0 

.80 

2 

July 

23 

0 

41 

21 

—  0 

.07 

,0 

Nov. 

20 

0 

18 

47 

—  0 

.21 

2 

1874 

Mar. 

20 

—  0 

6 

134 

+  ° 

.08 

2 

July 

18 

—  0 

29 

30 

+  ° 

.32 

2 

Nov. 

15 

—  0 

51 

0.00017 

+  o 

.07 

0  .0000 

To  sum  up :  A  Vp  and  Ar  are  each  very  small,  the 
former,  on  the  average,  somewhat  larger  than  the 
latter.  We  can  with  safety  neglect  further  consid- 
eration of  these  factors. 


A  sin  i  is  constant  and  of  sufficient  magnitude  to 
affect  A  in  the  hundredth^  of  second  for  several  (six) 
dates  about  the  time  of  perijovian  distance. 

There  remain  in  the  numerator  three  periodic 
terms, — namely,  sinu,  sin  ft'  and  h, — the  periods  of 
which  are,  respectively,  the  sidereal  period  of  Psyche 
(5.00  years),  the  sidereal  period  of  Jupiter  (11.86 
years),  and  the  synodic  period  of  Psyche  with  respect 
to  Jupiter  (8.64  years).  The  course  of  analysis  just 
employed  shows  that  none  of  these  has  a  controlling 
influence  in  determining  the  differences  between 
SCHUBERT'S  and  my  perturbations  in  n. 

By  following  out  exactly  similar  lines  of  investi- 


gation  for 


d 


ics 

and   ~j-  ,   which   it   seems   hardly 

Civ 


necessary  to  print,  it  became  certain  that  the  differ- 
ences between  SCHUBERT'S  elements  and  my  own  were 
quite  insufficient  to  produce  the  large  differences 
between  his  and  my  values  of  the  perturbations.  Much 
larger  increments  to  the  elements  would  be  necessary 
to  produce  such  large  differences  in  the  perturbations. 
It  was  then  thought  that  the  differences  might  have 
been  produced  by  the  use  of  different  values  of  the 
mass  of  Jupiter.  No  statement  was  made  by  SCHU- 
BERT of  the  value  of  the  mass  of  Jupiter  employed 
by  him,  but  an  examination  of  his  earlier  publications 
makes  it  very  probable  that  he  used  a  mass  factor 
quite  different  from  the  one  employed  by  me — 
BESSEL'S.  In'  publishing  general  perturbations  of 
Harmonia  by  Jupiter,  A.  N.  66,  p.  27,  December,  1865, 
SCHUBERT  makes  the  following  statements : — 

"  In  all  my  calculations  of  the  special  perturba- 
tions of  asteroids,  and  also  for  the  general  pertur- 
bations of  Melpomene  and  Eunomia,  the  mass  of 

Jupiter  by  BESSEL  1A._  QQ  has  been  used ;  but  as  I  have 

1U4:  (  .OO 


— 11 


been  led  to  think  the  mass  by  NICOLAI 


nearer 


the  truth,  the  above  perturbations  of  Harmonia  have 
been  computed  with  it.  On  former  occasions  I  have 
remarked  that  I  felt  certain  BESSEL  's  mass  of  Jupiter 
needed  a  correction,  and  that  it  would  be  possible  to 
determine  it  by  means  of  the  perturbations  of  Leu- 
kothca.  .  .  .  My  special  reasons  for  preferring  the 
mass  by  NICOLAI  to  that  by  BESSEL  are  the  following : 
In  the  great  many  definite  determinations  of  orbits  of 
asteroids  by  means  of  the  special  perturbations,  it 
would  appear  to  me  as  if  the  normals  could  be  repre- 
sented better  with  a  smaller  mass  of  Jupiter,  and, 
therefore,  for  the  sake  of  a  mere  trial,  once  in  the 
case  of  Thalia,  I  introduced  the  correction  as  the  sev- 

1  v 

onth  unknown  quantity  and  found  for  it  —  57^.    The 

oUUU 

nine  normals  upon  which  my  tables  of  Eunomia  are 
based  are  represented  better  with  NICOLAI 's  mass  than 
with  BESSEL 's.  The  same  result  has  been  arrived  at 
by  Professor  BRUNNOW  in  the  case  of  7m,  for  which 
he  has  computed  the  general  perturbations  by  Jupiter 
(first  and  second  order),  Saturn,  and  Mars." 

Assuming,   then,   that   SCHUBERT   used   NICOLAI 's 
value,    I   proceeded   to   compute   the    increments    to 


— r-  and   -    -  but  even  this  was  not  sufficient  to  ex- 
df  dt 

plain  away  the  large  differences  between  his  and 
my  perturbations.  In  fact,  if  these  increments  be 
plotted,  an  inspection  of  the  curves  shows  that  the 
differences  between  the  perturbations  increase  with 
the  time,  and  are  independent,  at  least  to  a  certain 
extent,  of  the  magnitude  of  the  perturbation.  This 
attempt  to  find  an  explanation  of  the  differences  be- 
tween the  two  sets  of  perturbations  has  been  made 
especially  difficult  by  the  fact  that  SCHUBERT  pub- 
lished no  details  whatever  in  regard  to  his  work, 
there  being  even  no  reference  to  the  formulae  and 
constants  used,  and  inquiry  at  the  Nautical  Almanac 
office  disclosed  the  fact  that  the  original  computation 
can  no  longer  be  found. 

There  remain  but  two  explanations  of  the  differ- 
ences: First,  that  SCHUBERT  did  not  correct  his  ele- 
ments from  time  to  time,  as  is  usually  done  in  the 
computation  of  special  perturbations  in  order  to  avoid 
the  accumulation  of  perturbations  of  the  second  order. 
Second,  that  either  one  or  both  of  the  computations 
is  affected  by  numerical  errors  of  a  constant  or  pro- 
gressive nature. 


As  to  the  first  of  these,  it  seems  hardly  possible 
that  such  an  experienced  computer  as  SCHUBERT 
should  have  done  this,  yet  an  inspection  of  my  com- 
putations shows  that  if  this  had  been  done,  the  result- 
ing perturbations  would  differ  from  those  obtained, 
in  the  same  way  that  SCHUBERT'S  do.  As  to  the 
second  hypothesis,  it  may  be  said  that  SCHUBERT  had 
had  long  experience  as  a  computer,  especially  in  the 
computation  of  general  perturbations,  but  I  believe 
Psyche  was  the  first  planet  for  which  he  computed 
special  perturbations.  I  have  every  confidence  in  the- 
correctness  of  my  own  computations.  Not  that  I 
never  make  mistakes, — alas!  I  have  learned  from 
bitter  experience  that  the  opposite  is  true, — but  that 
every  figure  of  the  computation  has  been  carefully 
checked,  and  parts  of  it  have  been  completely  and 
independently  recomputed  after  an  interval  of  two  or 
three  years. 

It  becomes  necessary  therefore  to  abandon  the 
attempt  to  explain  the  differences  between  SCHUBERT'S 
and  my  perturbations,  and  it  is  of  course  useless  to 
try  to  construct  empirical  formulas  by  which  to  carry 
the  corrections  forward  over  the  balance  of  the  thirty- 
year  period. 

Additional  Perturbations.  Instead  of  carrying 
out  the  second  part  of  this  plan  of  computation,  as 
stated  on  page  1,  I  expect  now  to  compute  general 
perturbations  for  Psyche  by  HANSEN'S  method.  While 
that  is  being  done,  however,  it  is .  desirable  to  keep 
track  of  the  planet,  and  I  have  therefore  computed 
special  perturbations  by  Jupiter  from  1900  to  1906, 
which  are  given  in  the  following  table.  They  have 
been  computed  from  my  corrected  elements  brought 
up  to  1900  by  means  of  SCHUBERT'S  perturbations, 
and  they  are  as  follows : — 

Epoch  1900  Jan.  0.    Berlin  M.  T. 

M  332°           1'  19" 

TT  16          34  55    \  Ecliptic  and 

O  150           31  44    \  mean  equinox 

»  3            4  31    )  of  1900.0. 

</>  7          51           3 

/*  710"  .4390 

The  constants,  elements,  and  coordinates  of  Jupiter 
have  been  taken  from  the  Berliner  Jahrbuch.  The 
elements  were  corrected  for  the  accumulated  pertur- 
bations whenever  it  was  necessary  to  do  so. 


—  12  — 


PERTURBATIONS  BY  JUPITER. 


Berlin  M.  T. 

/-1900 

di 

<m 

d<j> 

di 

r 

4* 

/«fc 

dM 

1900 

January   0.0 

0 

0".0 

0".0 

0".0 

0" 

.0 

0" 

.0000 

0".0 

0".0 

rBRAjV>x, 

February  9.0 

40 

-0  .1 

—  0  .6 

4-   6  .6 

—  21 

.1 

—  0 

.0425 

—  0  .8 

4-22  .2 

OF  THE 

March    21.0 

80 

0  .3 

1  .7 

13  .7 

40 

.1 

0 

.0868 

3  .4 

41  .0 

IVER3!TY 

April    30.0 

120 

0  .4 

3  .4 

21  .0 

56 

.0 

0 

.1314 

7  .8 

55  .3 

OF         /' 

June      9.0 

160 

0  .5 

5  .7 

28  .4 

68 

9 

0 

.1749 

13  .9 

64  .5 

J^QfiliW^ 

July     19.0 

200 

0  .6 

8  .6 

35  .5 

76 

.9 

0 

.2157 

21  .7 

69  .1 

August   28.0 

240 

0  .7 

11  .9 

42  .2 

82 

.8 

0 

.2528 

31  .1 

69  .8 

October   7.0 

280 

0  .7 

15  .7 

48  .5 

87 

.1 

0 

.2851 

41  .9 

67  .8 

November  16.0 

320 

0  .7 

19  .7 

54  .2 

91 

.1 

0 

.3119 

53  .9 

64  .6 

1900 

December  26.0 

360 

0  .7 

23  .9 

59  .5 

95 

.8 

0 

.3328 

66  .8 

61  .5 

1901 

February  4.0 

400 

0  .6 

28  .1 

64  .4 

102 

.4 

0 

.3478 

80  .4 

59  .6 

March    16.0 

440 

0  .5 

32  .2 

69  .0 

111 

.6 

0 

.3567 

94  .5 

59  .8 

April    25.0 

480 

0  .4 

36  .1 

73  .6 

123 

.7 

0 

.3598 

108  .8 

62  .7 

June     4.0 

520 

0  .2 

39  .7 

78  .1 

138 

.9 

0 

.3573 

123  .2 

68  .4 

July     14.0 

560 

—  0  .0 

42  .8 

82  .9 

156 

.9 

0 

.3495 

137  .3 

76  .7 

August   23.0 

600 

4-0  .2 

45  .4 

88  .0 

177 

.1 

0 

.3368 

151  .1 

87  .3 

October   2.0 

640 

0  .4 

47  .4 

93  .6 

198 

.8 

0 

.3195 

164  .2 

99  .5 

November  11.0 

680 

0  .7 

48  .9 

99  .6 

221 

.4 

0 

.2981 

176  .5 

112  .7 

1901 

December  21.0 

720 

0  .9 

49  .8 

106  .2 

243 

.7 

0 

.2730 

188  .0 

126  .0 

1902 

January  30.0 

760 

1  .2 

50  .0 

113  .5 

264 

.9 

0 

.2443 

198  .3 

138  .5 

March    11.0 

800 

1  .5 

49  .7 

121  .3 

284 

9 

0 

.2124 

207  .4 

149  .5 

April    20.0 

840 

1  .7 

49  .0 

129  .7 

300 

.7 

0 

.1776 

215  .2 

158  .1 

May     30.0 

880 

1  .9 

47  .7 

138  .6 

313 

.5 

0' 

.1403 

221  .6 

163  .7 

July      9.0 

920 

2  .2 

46  .1 

148  .0 

322 

.0 

0 

.1006 

226  .4 

165  .5 

August   18.0 

960 

2  .3 

44  .2 

157  .7 

325 

.5 

0 

.0588 

229  .6 

162  .9 

Septemb'r  27.0 

1000 

2  .5 

42  .0 

167  .8 

323 

.5 

—  0 

.0153 

231  .0 

155  .6 

November  6.0 

1040 

2  .7 

39  .8 

178  .0 

315 

.7 

4-  o 

.0300 

230  .7 

143  .2 

1902 

December  16.0 

1080 

2  .8 

37  .5 

188  .4 

301 

.8 

0 

.0767 

228  .6 

125  .4 

1903 

January  25.0 

1120 

2  .9 

35  .2 

198  .7 

281 

.8 

0 

.1245 

224  .6 

102  .4 

March    6.0 

1160 

2  .9 

33  .2 

208  .8 

255 

.6 

0 

.1732 

218  .6 

74  .1 

April    15.0 

1200 

3  .0 

31  .3 

218  .6 

223 

.6 

0 

9994 

210  .8 

40  .7 

May  .    25.0 

1240 

3  .0 

29  .7 

228  .1 

186 

.0 

0 

.2719 

200  .9 

4-  2  .9 

July      4.0 

1280 

3  .0 

28  .4 

237  .1 

143 

.5 

0 

.3212 

188  .9 

—  38  .9 

August   13.0 

1320 

3  .1 

27  .4 

245  .4 

97 

.0 

0 

.3699 

175  .1 

83  .8 

Septemb'r  22.0 

1360 

3  .1 

26  .8 

253  .1 

—  47 

.3 

0 

.4174 

159  .3 

130  .8 

November  1.0 

1400 

3  .1 

26  .5 

260  .1 

+  4 

.3 

0 

.4631 

141  .7 

178  .6 

1903 

December  11.0 

1440 

3  .1 

26  .5 

266  .3 

56 

.0 

0 

.5061 

122  .3 

225  .5 

1904 

January  20.0 

1480 

3  .1 

26  .7 

271  .8 

106 

.6 

0 

.5454 

101  .2 

270  .2 

February  29.0 

1520 

3  .1 

27  .1 

276  .6 

154 

.1 

0 

.5799 

78  .7 

310  .8 

April     9.0 

1560 

3  .1 

27  .5 

281  .0 

196 

.4 

0 

.6080 

54  .9 

345  .3 

May     19.0 

1600 

3  .1 

27  .8 

1'85  .1 

231 

.6 

0 

.6280 

30  .1 

371  .7 

June     28.0 

1640 

3  .1 

.  28  .1 

289  .2 

257 

.4 

0 

.6377 

4  .8 

388  .1 

August    7.0 

1680 

3  .1 

28  .1 

294  .0 

272 

.0 

0 

.6345 

4-  20  .7 

392  .8 

Septemb'r  16.0 

1720 

3  .1 

28  .0 

300  .0 

273 

.9 

0 

.6156 

45  .8 

384  .7 

October   26.0 

1760 

3  .1 

27  .8 

307  .8 

262 

9 

0 

.5775 

69  .7 

363  .:; 

1904 

December  5.0 

1800 

2  .9 

27  .7 

318  .5 

237 

.1 

0 

.5  1  ().l 

91  .7 

329  .5 

1905 

January  14.0 

1840 

2  .7 

28  .3 

332  .7 

199 

.9 

0 

.4292 

110  .7 

285  .5 

February  23.0 

1880 

2  .4 

30  .3 

351  .4 

153 

.3 

0 

.3129 

125  .7 

235  .3 

'  April     4.0 

1920 

2  .0 

35  .0 

375  .1 

100 

.6 

+  ° 

.1670 

135  .3 

183  .9 

.May      14.0 

1960 

1  .4 

43  .9 

403  .5 

4-  45 

.3 

-0 

.0050 

138  .7 

137  .1 

June     23.0 

2000 

4-  0  .7 

59  .3 

435  .6 

-  12 

.0 

0 

.1929 

134  .7 

97  .5 

August    2.0 

2040 

—  0  .0 

83  .0 

469  .0 

75 

.7 

0 

.3775 

123  .2 

62  .8 

Septemb'r  11.0 

2080 

0  .7 

116  .4 

500  .6 

157 

.8 

0 

.5314 

104  .9 

-  21  .4 

October   21.0 

2120 

1  .0 

158  .5 

526  .9 

275 

9 

(1 

.62;"(l 

SI  .(i 

4-45  .5 

1905 

November  30.0 

2160 

1  .0 

206  .1 

545  .8 

442 

.1 

0 

.6383 

56  .1 

157  .2 

1906 

January   9.0 

2200 

—  0  .5 

254  .3 

557  .7 

661 

.0 

0 

.5700 

31  .7 

322  .5 

February  18.0 

2240 

4-0  .3 

298  .4 

564  .6 

920 

9 

0 

.4376 

4-  1  1  .3 

534  .9 

1906 

March    30.0 

2280 

4-1  .4 

-335  .5 

4-569  .0 

-1199 

.3 

—  0 

.2674 

—  2  .8 

4-  776  .7 

. 


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